Modelling a sphere with holes on it
I think it is not necessary to combine these two models in the first place unless there is a specific reason why you would not model them separate.

Assuming there is a reason, I would do the following:


Start with a cube
Add and apply subdivision surface modifier level two
Shift-Alt-S + 1-Key in edit-mode to make the resulting shape a sphere
Shape the model according to you needs by insetting and extruding the corresponding faces.
Add subdivision surface modifier but don't apply this time unless you want to add even more detail the current topology does not cover.